Algebra 101 - Unit 8

Lesson 48: Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions requires a common denominator. Once the denominators are the same, you combine the numerators and simplify.

Steps for Adding and Subtracting

1. Factor all denominators completely.
2. Find the least common denominator (LCD).
3. Rewrite each fraction with the LCD as the denominator.
4. Add or subtract the numerators.
5. Simplify the resulting rational expression.

Examples

Example 1: Add 1/(x + 2) + 3/(x − 1)

• LCD = (x + 2)(x − 1)
• Rewrite fractions: (1)(x − 1)/(x + 2)(x − 1) + (3)(x + 2)/(x + 2)(x − 1)
• Add numerators: (x − 1 + 3x + 6)/(x + 2)(x − 1)
• Simplify numerator: (4x + 5)/(x + 2)(x − 1)

Example 2: Subtract (x² + x)/(x² − 1) − (2x)/(x + 1)

• Factor denominators: x² − 1 = (x − 1)(x + 1)
• LCD = (x − 1)(x + 1)
• Rewrite fractions: (x² + x)/(x − 1)(x + 1) − (2x)(x − 1)/(x − 1)(x + 1)
• Subtract numerators: (x² + x − 2x² + 2x)/(x − 1)(x + 1)
• Simplify numerator: (−x² + 3x)/(x − 1)(x + 1) → Factor: −x(x − 3)/(x − 1)(x + 1)

Practice Problems

1. Add: 1/(x − 2) + 2/(x + 3)
2. Subtract: (x + 1)/(x² − 9) − (2)/(x − 3)
3. Add: (x²)/(x² − 4) + (3)/(x + 2)
4. Subtract: (2x)/(x² − x − 6) − (x + 1)/(x − 3)
5. Add: 3/(x + 1) + 5/(x − 2)